Theorem axhilex 10483

Description: Derive axiom ax-hilex 10501 from Hilbert space under ZF set theory.

Before introducing the 18 axioms for Hilbert space, we first prove them as the conclusions of theorems axhilex 10483 through axhcompl 10500, using ZFC set theory only. These show that if we are given a known, fixed Hilbert space U = <.<. +h , .h >., normh>. that satisfies their hypotheses, then we can derive the Hilbert space axioms as theorems of ZFC set theory. In practice, in order to use these theorems to convert the Hilbert Space explorer to a ZFC-only subtheory, we would also have to provide definitions for the 3 (otherwise primitive) class constants +h, .h, and .ih before df-hnorm 10469 above. See also the comment in ax-hilex 10501.

Hypotheses

Ref	Expression
axhil.1	|- U = <.<. +h , .h >., normh>.
axhil.2	|- U e. CHil

Assertion

Ref	Expression
axhilex	|- ~H e. _V

Proof of Theorem axhilex

Step	Hyp	Ref	Expression
1		df-hba 10470	. 2 |- ~H = (BaseSet` <.<. +h , .h >., normh>.)
2	1	hlex 9947	1 |- ~H e. _V

Syntax hints:   = wceq 1298   e. wcel 1300  _Vcvv 2292  <.cop 3046  CHilchl 9934  ~Hchil 10420   +h cva 10421   .h csm 10422  normhcno 10426

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-uni 3178  df-fv 4014  df-hba 10470

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